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19
A Fast MultiScale Method for Drawing Large Graphs
 JOURNAL OF GRAPH ALGORITHMS AND APPLICATIONS
, 2002
"... We present a multiscale layout algorithm for the aesthetic drawing of undirected graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that are substantially larger than those we have encountered in prior work. For example, the paper contains a drawi ..."
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Cited by 79 (10 self)
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We present a multiscale layout algorithm for the aesthetic drawing of undirected graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that are substantially larger than those we have encountered in prior work. For example, the paper contains a drawing of a graph with over 15,000 vertices. Also we achieve "nice" drawings of 1000 vertex graphs in about 1 second. The proposed algorithm embodies a new multiscale scheme for drawing graphs, which was motivated by the earlier multiscale algorithm of Hadany and Harel [HH99]. In principle, it could significantly improve the speed of essentially any forcedirected method (regardless of that method's ability of drawing weighted graphs or the continuity of its costfunction).
Graph Drawing by HighDimensional Embedding
 In GD02, LNCS
, 2002
"... We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seco ..."
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Cited by 59 (10 self)
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We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.
An Energy Model for Visual Graph Clustering
 Proceedings of the 11th International Symposium on Graph Drawing (GD 2003), LNCS 2912
, 2003
"... We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the bestknown force and energy models do not clearly show clusters for graphs whose ..."
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Cited by 41 (4 self)
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We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the bestknown force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawings of our energy model. This characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted.
Drawing Graphs with NonUniform Vertices
, 2002
"... The vertices of most graphs that appear in real applications are nonuniform. They can be circles, ellipses, rectangles, or other geometric elements of varying shapes and sizes. Unfortunately, current force directed methods for laying out graphs are suitable mostly for graphs whose vertices are zero ..."
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Cited by 29 (3 self)
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The vertices of most graphs that appear in real applications are nonuniform. They can be circles, ellipses, rectangles, or other geometric elements of varying shapes and sizes. Unfortunately, current force directed methods for laying out graphs are suitable mostly for graphs whose vertices are zerosized and dimensionless points. It turns out that naively extending these methods to handle nonuniform vertices results in serious deficiencies in terms of output quality and performance. In this paper we try to remedy this situation by identifying the special characteristics and problematics of such graphs and offering several algorithms for tackling them. The algorithms can be viewed as carefully constructed extensions of forcedirected methods, and their output quality and performance are similar.
Visual Clustering of Graphs with Nonuniform Degrees
 Proceedings of the 13th International Symposium on Graph Drawing (GD 2005
, 2004
"... We discuss several criteria for clustering graphs, and identify two criteria which are not biased towards certain cluster sizes: the nodenormalized cut (also called cut ratio) and the edgenormalized cut. We present two energy models whose minimum energy drawings reveal clusters with respect to ..."
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Cited by 27 (2 self)
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We discuss several criteria for clustering graphs, and identify two criteria which are not biased towards certain cluster sizes: the nodenormalized cut (also called cut ratio) and the edgenormalized cut. We present two energy models whose minimum energy drawings reveal clusters with respect to these criteria.
Energy Models for Drawing Clustered SmallWorld Graphs
, 2003
"... We introduce energy models for drawing clustered smallworld graphs. ..."
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Cited by 8 (3 self)
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We introduce energy models for drawing clustered smallworld graphs.
A MaxentStress Model for Graph Layout
"... In some applications of graph visualization, input edges have associated target lengths. Dealing with these lengths is a challenge, especially for large graphs. Stress models are often employed in this situation. However, the traditional full stress model is not scalable due to its reliance on an in ..."
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Cited by 4 (2 self)
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In some applications of graph visualization, input edges have associated target lengths. Dealing with these lengths is a challenge, especially for large graphs. Stress models are often employed in this situation. However, the traditional full stress model is not scalable due to its reliance on an initial allpairs shortest path calculation. A number of fast approximation algorithms have been proposed. While they work well for some graphs, the results are less satisfactory on graphs of intrinsically high dimension, because nodes overlap unnecessarily. We propose a solution, called the maxentstress model, which applies the principle of maximum entropy to cope with the extra degrees of freedom. We describe a forceaugmented stress majorization algorithm that solves the maxentstress model. Numerical results show that the algorithm scales well, and provides acceptable layouts for large, nonrigid graphs. This also has potential applications to scalable algorithms for statistical multidimensional scaling (MDS) with variable distances.
One Dimensional Layout Optimization, with Applications to Graph Drawing by Axis Separation
, 2005
"... In this paper we discuss a useful family of graph drawing algorithms, characterized by their ability to draw graphs in one dimension. We define the special requirements from such algorithms and show how several graph drawing techniques can be extended to handle this task. In particular, we suggest ..."
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Cited by 1 (1 self)
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In this paper we discuss a useful family of graph drawing algorithms, characterized by their ability to draw graphs in one dimension. We define the special requirements from such algorithms and show how several graph drawing techniques can be extended to handle this task. In particular, we suggest a novel optimization algorithm that facilitates using the Kamada and Kawai model [17] for producing onedimensional layouts. The most important application of the algorithms seems to be in achieving graph drawing by axis separation, where each axis of the drawing addresses different aspects of aesthetics.
Drawing
"... We propose a graph drawing algorithm that is both efficient and high quality. This algorithm combines a multilevel approach, which effectively overcomes local minimums, with the Barnes and Hut [1] octree technique, which approximates short and longrange force efficiently. Our numerical results sho ..."
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We propose a graph drawing algorithm that is both efficient and high quality. This algorithm combines a multilevel approach, which effectively overcomes local minimums, with the Barnes and Hut [1] octree technique, which approximates short and longrange force efficiently. Our numerical results show that the algorithm is comparable in speed to Walshaw’s [2] highly efficient multilevel graph drawing algorithm, yet gives better results on some of the difficult problems. In addition, an adaptive cooling scheme for the forcedirected algorithms and a general repulsive force model are proposed. The proposed graph drawing algorithm and others are included with Mathematica 5.1 and later versions in the package DiscreteMath‘GraphÑ Plot.
Drawing Large Graphs by LowRank Stress Majorization
"... Optimizing a stress model is a natural technique for drawing graphs: one seeks an embedding into R d which best preserves the induced graph metric. Current approaches to solving the stress model for a graph with V  nodes and E  edges require the full allpairs shortest paths (APSP) matrix, which ..."
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Optimizing a stress model is a natural technique for drawing graphs: one seeks an embedding into R d which best preserves the induced graph metric. Current approaches to solving the stress model for a graph with V  nodes and E  edges require the full allpairs shortest paths (APSP) matrix, which takes O(V  2 logE  + VE) time and O(V  2) space. We propose a novel algorithm based on a lowrank approximation to the required matrices. The crux of our technique is an observation that it is possible to approximate the full APSP matrix, even when only a small subset of its entries are known. Our algorithm takes time O(kV  + VlogV  + E) per iteration with a preprocessing time of O(k 3 + k(E  + VlogV) + k 2 V) and memory usage of O(kV), where a userdefined parameter k trades off quality of approximation with running time and space. We give experimental results which show, to the best of our knowledge, the largest (albeit approximate) full stress model based layouts to date.